IV. Supplementary researches on the partition of numbers

The general formula given at the conclusion of my memoir, “Researches on the Partition of Numbers,” is somewhat different from the corresponding formula of Professor Sylvester, and leads more directly to the actual expression for the number of partitions, in the form made use of in my memoir; to complete my former researches, I propose to explain the mode of obtaining from the formula the expression for the number of partitions. The formula referred to is as follows, viz. ifφx /fx be a rational fraction, the denominator of which is made up of factors (the same or different) of the form 1 —xm , and ifa is a divisor of one or more of the indicesm , andk is the number of indices of which it is a divisor, then {φx /fx }[1 -xα ]= . . . + 1/π(s - 1)(x∂x )s - 1Sχς /ς-x ... = . . . + 1/π(s - 1)(x∂x )s - 1θx /[1-xα ]... whereχς = coeff. 1/t int s -1ςφ (ςe-t )/f (ςe-t ), in which formula [1-xn ] denotes the irreducible factor of 1-xn , that is, the factor which equated to zero gives the prime roots, andς is a root of the equation [1-xn ]=0; the summation of course extends to all the roots of the equation. The indexs extends froms = 1 tos =k ; and we have then the portion of the fraction depending on the denominator [1-xn ].